Oscillatory behaviour of a higher-order dynamic equation

The calculus on time scales has been introduced in order to unify the theories of continuous and discrete processes and in order to extend those theories to a more general class of the so-called dynamic equations. In recent years there has been much research activity concerning the oscillation and non-oscillation of solutions of neutral dynamic equations on time scales.

where $n\ge 2$, $P(t),Q_i(t)\in C_{rd}{[t_0,\mathrm{\infty })}_{\mathbb{T}}$ for $i=1,2,\dots ,m$; $P(t)$ is an oscillating function ($P(t):\mathbb{T}\to \mathbb{R}$), $Q_i(t)$ are positive real-valued functions for $i=1,2,\dots ,m$; ${\varphi }_i(t)\in C_{rd}{[t_0,\mathrm{\infty })}_{\mathbb{T}}$, ${\varphi }_i^{\mathrm{\Delta }}(t)>0$, the variable delays $\tau ,{\varphi }_i:{[t_0,\mathrm{\infty })}_{\mathbb{T}}\to \mathbb{T}$ with $\tau (t),{\varphi }_i(t)<t$ for all $t\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$, ${\varphi }_i(t)\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$ for $i=1,2,\dots ,m$; $\tau (t)\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$; $f_i(u)\in C(\mathbb{R},\mathbb{R})$ are nondecreasing functions, $u{f}_i(u)>0$ for $u\ne 0$ and $i=1,2,\dots ,m$.

We introduce the set ${\mathbb{T}}^{\kappa }$ which is derived from the time scale $\mathbb{T}$ as follows. If $\mathbb{T}$ has left-scattered maximum m, then ${\mathbb{T}}^{\kappa }=\mathbb{T}-\{m\}$, otherwise ${\mathbb{T}}^{\kappa }=\mathbb{T}$.

Definition 1 [1]

The function $f:\mathbb{T}\to \mathbb{R}$ is called rd-continuous provided it is continuous at right-dense points in $\mathbb{T}$ and its left-sided limits exist (finite) at left-dense points in $\mathbb{T}$.

Theorem 1 [1]

where we denote the derivative on $\tilde{\mathbb{T}}$ by $\tilde{\mathrm{\Delta }}$.

Definition 2 [1]

Theorem 2 [2]

is nonempty, and so is $L_{\epsilon }(\mathrm{\infty })=\{t\in \mathbb{T}:t>\frac{1}{\epsilon }\}$ if $\mathrm{\infty }\in \tilde{\mathbb{T}}$.

Lemma 1 [3]

Lemma 2 [3]

Lemma 3 [3]

Lemma 4 [3]

and therefore the proof of the lemma can be immediately completed. □

For the cases $\mathbb{T}=\mathbb{Z}$ and $\mathbb{T}=\mathbb{R}$, some sufficient criterias for oscillatory behaviour of the solutions of the equation (1.1) were obtained by Bolat and Akın in [6] and [7], respectively. Furthermore, there might be other time scales that we cannot appreciate at this time due to our current lack of ‘real-world’ examples.

Theorem 3 Assume that n is odd and

(C1) ${lim}_{t\to \mathrm{\infty }}P(t)=0$,

(C2) ${\int }_{t_0}^{\mathrm{\infty }}{s}^{n-1}{\sum }_{i=1}^m{Q}_i(s)\mathrm{\Delta }s=\mathrm{\infty }$.

Then every bounded solution of equation (1.1) is either oscillatory or tends to zero as $t\to \mathrm{\infty }$.

It follows that $z^{{\mathrm{\Delta }}^j}(t)$ ($j\in [0,n-1]\cap {\mathbb{N}}_0$) is strictly monotone and eventually of constant sign. Since $P(t)$ is an oscillatory function, there exists a $t_2\ge t_1$ such that if $t\ge t_2$, then $z(t)>0$. Since $y(t)$ is bounded, by virtue of (C1) and (1.2), there is a $t_3\ge t_2$ such that $z(t)$ is also bounded for $t\ge t_3$. Because n is odd and $z(t)$ is bounded, by Lemma 1, when $p=0$ (otherwise $z(t)$ is not bounded), there exists $t_4\ge t_3$ such that for $t\ge t_4$ we have ${(-1)}^j{z}^{{\mathrm{\Delta }}^j}(t)>0$, $j\in [0,n-1]\cap {\mathbb{N}}_0$.

from (1.2).

for $t\ge t_1$. That is, $z^{{\mathrm{\Delta }}^n}>0$. It follows that $z^{{\mathrm{\Delta }}^j}(t)$ ($j\in [0,n-1]\cap {\mathbb{N}}_0$) is strictly monotone and eventually of constant sign. Since $P(t)$ is an oscillatory function, there exists a $t_2\ge t_1$ such that if $t\ge t_2$, then $z(t)<0$. Since $y(t)$ is bounded, by (C1) and (1.2), there is a $t_3\ge t_2$ such that $z(t)$ is also bounded for $t\ge t_3$. Assume that $x(t)=-z(t)$. Then $x^{{\mathrm{\Delta }}^n}(t)=-z^{{\mathrm{\Delta }}^n}(t)$. Therefore, $x(t)>0$ and $x^{{\mathrm{\Delta }}^n}(t)<0$ for $t\ge t_3$. Hence, we observe that $x(t)$ is bounded. Since n is odd, by Lemma 1, there exists a $t_4\ge t_3$ and $p=0$ (otherwise $x(t)$ is not bounded) such that ${(-1)}^j{x}^{{\mathrm{\Delta }}^j}(t)>0$, $j\in [0,n-1]\cap {\mathbb{N}}_0$ and $t\ge t_4$. That is, ${(-1)}^j{z}^{{\mathrm{\Delta }}^j}(t)<0$, $j\in [0,n-1]\cap {\mathbb{N}}_0$ and $t\ge t_4$. In particular, for $t\ge t_4$ we have $z^{\mathrm{\Delta }}(t)>0$. Therefore, $z(t)$ is increasing. So, we can assume that ${lim}_{t\to \mathrm{\infty }}z(t)=L$ ($-\mathrm{\infty }<L\le 0$). As in the proof of $y(t)>0$, we may prove that $L=0$. As for the rest, it is similar to the case of $y(t)>0$. That is, ${lim}_{t\to \mathrm{\infty }}y(t)=0$. This contradicts our assumption. Hence the proof is completed. □

Theorem 4 Assume that n is even and (C 1) holds. If the following condition is satisfied:

for $\phi (t)$ and $i=1,2,\dots ,m$. Then every bounded solution of equation (1.1) is oscillatory.

This is a contradiction.

for $t\ge t_1$. That is, $z^{{\mathrm{\Delta }}^n}>0$. It follows that $z^{{\mathrm{\Delta }}^j}(t)$ ($j\in [0,n-1]\cap {\mathbb{N}}_0$) is strictly monotone and eventually of constant sign. Since $P(t)$ is an oscillatory function, there exists a $t_2\ge t_1$ such that $z(t)<0$ for $t\ge t_2$. Since $y(t)$ is bounded, by (C1) and (1.2), there is a $t_3\ge t_2$ such that $z(t)$ is also bounded for $t\ge t_3$. Assume that $x(t)=-z(t)$. Then $x^{{\mathrm{\Delta }}^n}(t)=-z^{{\mathrm{\Delta }}^n}(t)$. Therefore, $x(t)>0$ and $x^{{\mathrm{\Delta }}^n}(t)<0$ for $t\ge t_3$. Hence, we observe that $x(t)$ is bounded. Since n is odd, by Lemma 1, there exists a $t_4\ge t_3$ and $p=1$ (otherwise $x(t)$ is not bounded) such that ${(-1)}^k{x}^{{\mathrm{\Delta }}^k}(t)>0$, $k\in [0,n-1]\cap {\mathbb{N}}_0$ and $t\ge t_4$. That is, ${(-1)}^k{z}^{{\mathrm{\Delta }}^k}(t)<0$, $k\in [0,n-1]\cap {\mathbb{N}}_0$ and $t\ge t_4$. In particular, for $t\ge t_4$ we have $z^{\mathrm{\Delta }}(t)>0$. Therefore, $z(t)$ is increasing. For the rest of the proof, we can proceed the proof similarly to the case of $y(t)>0$. Hence, the proof is completed. □